The Geometry of Efficient Fair Division

Finding maxmin allocations in cooperative and competitive fair division

“…By (RD) we have that for any point on Pareto border of P there exists one and only one hyperplane touching P (see [2]). By the conclusions of Proposition 4.5 either α t * ∈ ∆ * m−1 or lim t→+∞ α t = α * ∈ ∆ * m−1 .…”

“…We now describe a geometrical setting already employed in [1], [2], [7] and [14] to explore fair division problems. In what follows we consider the weighted preferences and densities, µ w j and f w j , given respectively by…”

“…The PVV u α is a point where the hyperplane m j=1 α j x j = k touches the partition range P, so u α lies on the Pareto border of P. Moreover, for any α ∈ ∆ m−1 there exists at least one PVV (see [2]). We are ready to state the first approximation result.…”

“…In Section 2 we describe the maxmin fair division problem with coalitions through the strategic model of interaction among players in [8] and the geometrical setting employed in [1], [2] and [7]. In Section 3 we present the upper and the lower bounds for the objective value.…”

N-Person Cake-Cutting: There May Be No Perfect Division

“…Books giving overviews of both existence results and algorithms for physically cutting a cake include Brams and Taylor [11], Robertson and Webb [17], Barbanel [1], and Brams [7]. Review articles of fair division that discuss cake-cutting include Brams [6] and Klamler [16].…”